Classical optimizer for quantum chemistry circuit synthesis

ABSTRACT

A computer-implemented method, system and a computer readable medium storing executable instructions for optimizing a quantum circuit are disclosed. The computer-implemented method includes receiving one or more parameters for simulation of evolution of at least one quantum state of a chemical entity to be simulated; generating a quantum circuit for the simulation; performing one or more operations to minimize quantum resources to be used for the generated quantum circuit based on the one or more parameters; and placing quantum resources among one or more elementary logical units (ELUs) based on any one or more of: frequency of occurrence of the quantum resources in the generated quantum circuit, order of occurrence of the quantum resources in the generated quantum circuit, connectivity parameters between one or more quantum resources, efficiency of gates between specific quantum resources, quality of gates between specific quantum resources or a combination thereof.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. application Ser. No.16/213,379, filed Dec. 7, 2018, which is incorporated herein byreference by its entirety.

TECHNICAL FIELD

The embodiments described herein pertain generally to generatingoptimized quantum circuits for simulation of quantum system for achemical entity.

BACKGROUND

A quantum computer is a computational system which usesquantum-mechanical phenomena, such as superposition and entanglement, toprocess data. Unlike digital computers, in which data is encoded intobinary digits (bits) in one of two definite states (“0” or “1”), quantumcomputation requires data to be encoded into quantum bits (hereafter“qubits”), for which a single qubit may represent a “1,” a “0,” or anyquantum superposition of the two qubit states. In general, a quantumcomputer with N qubits may be in an arbitrary superposition of up to 2″different states simultaneously, i.e., a pair of qubits may be in anyquantum superposition of four states, and three qubits may be in anysuperposition of eight states.

Quantum computers are able to solve certain problems much more quicklyand efficiently than digital computers (alternatively referred to hereinas “classical computers”). In the operation of a quantum computer,computations may be initialized by setting qubits in a controlledinitial state. By manipulating those qubits, predetermined sequences ofquantum logic gates are realized that represent the solution to aproblem to be solved, called a quantum algorithm. Quantum algorithms,such as Shor's algorithm, Simon's algorithm, etc., run faster than anypossible probabilistic classical algorithm.

Based on the inherent advantages in quantum computers in solving certainproblems, the challenge is in programming quantum computers to takeadvantage of their strengths in solving those problems.

SUMMARY

In one example embodiment, a computer-implemented method for optimizinga quantum circuit includes receiving one or more parameters forsimulation of evolution of at least one quantum state of a chemicalentity to be simulated; generating a quantum circuit for simulation ofevolution of the at least one quantum state of the chemical entity;performing one or more operations to minimize quantum resources to beused for the generated quantum circuit based on the one or moreparameters; and placing quantum resources among one or more elementarylogical units (ELUs) based on any one or more of: frequency ofoccurrence of the quantum resources in the generated quantum circuit,order of occurrence of the quantum resources in the generated quantumcircuit, connectivity parameters between one or more quantum resources,efficiency of gates between specific quantum resources, quality of gatesbetween specific quantum resources or a combination thereof.

In another example embodiment, a system for optimizing a quantum circuitcomprising at least one processor and a memory wherein the memory storesexecutable instructions for optimizing a quantum circuit that, uponexecution by the processor, cause the processor to perform functionsincluding receiving one or more parameters for simulation of evolutionof at least one quantum state of a chemical entity to be simulated;generating a quantum circuit for simulation of evolution of the at leastone quantum state of the chemical entity; performing one or moreoperations to minimize quantum resources to be used for the generatedquantum circuit based on the one or more parameters; and placing quantumresources among one or more elementary logical units (ELUs) based on anyone or more of: frequency of occurrence of the quantum resources in thegenerated quantum circuit, order of occurrence of the quantum resourcesin the generated quantum circuit, connectivity parameters between one ormore quantum resources, efficiency of gates between specific quantumresources, quality of gates between specific quantum resources or acombination thereof.

In yet another embodiment, a computer-readable medium storing executableinstructions for optimizing a quantum circuit that, upon execution,cause a digital computing processor to perform functions includingreceiving one or more parameters for simulation of evolution of at leastone quantum state of a chemical entity to be simulated; generating aquantum circuit for simulation of evolution of the at least one quantumstate of the chemical entity; performing one or more operations tominimize quantum resources to be used for the generated quantum circuitbased on the one or more parameters; and placing quantum resources amongone or more elementary logical units (ELUs) based on any one or more of:frequency of occurrence of the quantum resources in the generatedquantum circuit, order of occurrence of the quantum resources in thegenerated quantum circuit, connectivity parameters between one or morequantum resources, efficiency of gates between specific quantumresources, quality of gates between specific quantum resources or acombination thereof.

The foregoing summary is illustrative only and is not intended to be inany way limiting. In addition to the illustrative aspects, embodiments,and features described above, further aspects, embodiments, and featureswill become apparent by reference to the drawings and the followingdetailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

In the detailed description that follows, embodiments are described asillustrations only since various changes and modifications will becomeapparent to those skilled in the art from the following detaileddescription. The use of the same reference numbers in different figuresindicates similar or identical items.

FIG. 1A illustrates a portion of an exemplary quantum systemarchitecture in accordance with at least some embodiments describedherein;

FIG. 1B illustrates an exemplary elementary logic unit of the quantumsystem architecture in accordance with at least some embodimentsdescribed herein;

FIGS. 2A and 2B show an example processing flow by which at leastportions of the method for optimizing a quantum circuit are implemented,in accordance with at least some embodiments described herein;

FIGS. 3A and 3B show different methods for performing gate operationsusing quantum circuit, in accordance with at least some embodimentsdescribed herein;

FIGS. 4A, 4B and 4C show an example processing flow by which at leastportions of the method for optimizing a quantum circuit are implemented,in accordance with at least some embodiments described herein;

FIG. 4D shows an example processing flow by which at least portions ofthe method for optimizing a quantum circuit are implemented, inaccordance with at least some embodiments described herein;

FIG. 4E shows an example processing flow by which at least portions ofthe method for optimizing a quantum circuit are implemented, inaccordance with at least some embodiments described herein;

FIG. 4F shows an example processing flow by which at least portions ofthe method for optimizing a quantum circuit are implemented, inaccordance with at least some embodiments described herein;

FIG. 5 shows a sample circuit created by at least portions of thedifferent embodiments of the method for optimizing a quantum circuit, inaccordance with at least some embodiments described herein;

FIG. 6 Illustrates an example quantum computing system used to providequantum state measurements for a chemical entity chose for which thequantum circuit is optimized using method and system for optimization ofquantum circuit according to one or more embodiments described herein;and

FIG. 7 shows an example classical computing system including an examplecomputing embodiment, in which any of the processes and sub-processesfor optimization of quantum circuits may be implemented ascomputer-readable instructions stored on a computer-readable medium.

DETAILED DESCRIPTION

In the following detailed description, reference is made to theaccompanying drawings, which form a part of the description. In thedrawings, similar symbols typically identify similar components, unlesscontext dictates otherwise. Furthermore, unless otherwise noted, thedescription of each successive drawing may reference features from oneor more of the previous drawings to provide clearer context and a moresubstantive explanation of the current example embodiment. Still, theexample embodiments described in the detailed description, drawings, andclaims are not meant to be limiting. Other embodiments may be utilized,and other changes may be made, without departing from the spirit orscope of the subject matter presented herein. It will be readilyunderstood that the aspects of the present disclosure, as generallydescribed herein and illustrated in the drawings, may be arranged,substituted, combined, separated, and designed in a wide variety ofdifferent configurations, all of which are explicitly contemplatedherein.

Described herein are approaches embodied in one or more of systems,apparatuses, applications, programs, and methods by which quantum hybridcomputation including both classical and quantum computing are securelyand efficiently implemented by, e.g., apportioning computing betweenco-located computing devices in disparate computing environments.

In the present description, the following terms may be used, in additionto their accepted meaning, as follows:

“Classical computing,” “classical program,” “digital computing,”“digital program,” or variations thereof may refer tocomputing/processing of data that has been converted into binarynumbers. Classical computing processors may include, but not be limitedto, a central processing unit (CPU), a graphical processing unit (GPU),a tensor processing unit (TPU), application-specific integrated circuits(ASICs), field programmable gate arrays, etc., and unbound permutationsthereof. Non-limiting examples thereof may include a classicaloptimization function that may be parallelized and run on a multicoreclassical computer; a classical computer with a GPU for performingparallel calculations, etc.

“Quantum computing,” or variations thereof, may refer tocomputing/processing of data that has been encoded into qubits. Quantumcomputing utilizes qubits to perform high-level gating functions toproduce results that are faster than classical computing for a specificclass of computational problems.

“Environment” may refer to a computing environment in which are foundcomponents that when executing program may utilize, e.g., utilities suchas libraries, other programs, other hardware, etc. Thus, reference maybe made herein to a classical computing environment, a quantum computingenvironment, etc.

A “circuit” may refer to one or more quantum functions.

In accordance with the example embodiments described herein, quantumcomputing includes executing iterative processes by which a quantumcircuit may be written in a classical computing environment forexecution in a quantum computing environment. Non-limiting examples ofsuch classical computing environment that can be used include a desktopcomputer, a laptop computer, mobile device, etc. The quantum circuit maybe submitted via a network, e.g., the Internet, to a quantum computingdevice at which the quantum circuit may be queued with other quantumcircuits, in the same manner as, for example, batch processing for amainframe computing device. The queued quantum circuits may be executedin turn.

Iterative, or cumulative, results of the computing of the quantumcircuit may be received and collated in either the classical computingenvironment in which the quantum circuit was written or in a differentclassical computing environment in which the execution of the quantumcircuit is managed. Whichever classical computing environment, thequantum circuit may be updated or rewritten based upon results of themost recent iteration or upon cumulative results of the computing up tothat point.

Quantum chemistry problems are considered to be one of importantapplications of quantum computers. The embodiments described hereindisclose a method, a system and a computer program product to synthesizeefficient quantum circuits that are used to simulate an atom, a moleculeand/or an ion evolving according to its particular evolution operator.The evolution operator at a given time e^(iHt) may be derived by using atime dependent or time independent Schrôdinger's Equation. Theembodiments described herein improve the efficiency of quantum circuits,maximize savings of quantum resources and leverage unique capabilities,such as global gates that offer a more efficient implementation whencompared to efficiency offered by usage of local entangling gates. Theglobal gates may be provided by, e.g., ion-trap quantum computers, tofurther save quantum resources. Other computers that provide globalgates may also be used. Quantum resources are defined as quantum bitsalso known as qubits required for computations.

FIG. 1A illustrates a portion of an exemplary quantum systemarchitecture in accordance with at least some embodiments describedherein. As illustrated in FIG. 1A, the exemplary system architectureincludes a number of extended elementary logic units (EELUs) 110 ¹-110^(n), 112 ¹-112 ^(n), 114 ¹-114 ^(n) . . . etc. Each EELU furtherincludes a number of N-qubit registers, also known as elementary logicunits (ELUs) 120 ¹-120 ^(n), 122 ¹-122 ^(n), 124 ¹-124 ^(n) . . . etc.,which further include trapped ion qubits as shown in FIG. 1B.Controlling the states of these qubits by applying adequate controlsignals, for example, via laser beam, to one or more subsets of thesequbits, quantum gates between the subset of qubits may be realized. Thequbits from different EELUs and ELUs are interconnected or entangledthrough photonic quantum channels 126 ¹-126 ^(x) using reconfigurableoptical or photonic switch 128. In this approach, a pair of entangledqubits can be generated between two EELUs, where one qubit from the pairresides on each EELU being connected. Such an entangled pair can begenerated by an ion from each EELU emitting a single photon whoseinternal degree of freedom (such as the polarization, frequency, or aposition in time) serves as a photonic qubit that is entangled with theion that is emitting this photon. Two photons emitted by each ion in thetwo EELUs are brought together at a 50/50 beam splitter 132 ¹ . . . 132^(x/2), and subsequently detected at the two output ports 134 ¹ . . .134 ^(x) of the beam splitter. When a specific outcome is achieved (suchas each output port of the beam splitter detecting one photonsimultaneously), this signals a successful creation of an entangled ionpair between the EELUs. Use of an optical crossconnect switch (or aphotonic switch) 128 allows photons from arbitrary pairs of the EELUs inthe system to be brought together via photonic quantum channels 126 ¹ .. . 126 ^(x), enabling the generation of entangled ion pairs between anarbitrary pair of EELUs in the system. Once the entangled pair isgenerated, it can be used as a resource to either teleport a qubit fromone of the EELUs to the other (known as quantum teleportation process),or apply two-qubit entangling gate between an ion qubit in one EELU andanother ion qubit in the other EELU (known as teleporting a gate).

FIG. 1B illustrates an exemplary elementary logic unit of the quantumsystem architecture in accordance with at least some embodimentsdescribed herein. As illustrated in FIG. 1A, exemplary systemarchitecture includes a number of extended elementary logic units(EELUs), for example, 110. As illustrated in FIG. 1B, each EELU furtherincludes a number of registers, also known as elementary logic units(ELUs) 120 ¹-120 ^(n), which further include trapped ion qubits 130¹-130 ^(P) referred to as trapped ions or qubits and also as quantumresources. Controlling the states of these qubits 130 ¹-130 ^(P) byapplying adequate control signal, for example, via laser beam, to one ormore subsets of these qubits 130 ¹-130 ^(P) quantum gates between thesubset of qubits 130 ¹-130 ^(P) may be realized via local Coulombinteractions between qubits 140. The number of trapped ions in an ELUmay change over time, as trapped ions are shuttled from one ELU toanother to realize quantum functions.

Details of the scalable quantum system architecture described in FIGS.1A and 1B are provided in U.S. Pat. No. 9,858,531 entitled “FAULTTOLERANT SCALABLE MODULAR QUANTUM COMPUTER ARCHITECTURE WITH AN ENHANCEDCONTROL OF MULTI-MODE COUPLINGS BETWEEN TRAPPED ION QUBITS”.

In one example embodiment, a computer-implemented method for optimizinga quantum circuit includes receiving one or more parameters forsimulation of evolution of at least one quantum state of a chemicalentity to be simulated; generating a quantum circuit for simulation ofevolution of the at least one quantum state of the chemical entity;performing one or more operations to minimize quantum resources to beused for the generated quantum circuit based on the one or moreparameters; and placing quantum resources among one or more elementarylogical units (ELUs) based on any one or more of: frequency ofoccurrence of the quantum resources in the generated quantum circuit,order of occurrence of the quantum resources in the generated quantumcircuit, connectivity parameters between one or more quantum resources,efficiency of gates between specific quantum resources, quality of gatesbetween specific quantum resources or a combination thereof.

The computer-implemented method for optimizing a quantum circuitincludes receiving one or more parameters for simulation of evolution ofat least one quantum state of a chemical entity to be simulated;deriving a time evolution operator e^(iHt) for the chemical entity,wherein deriving evolution operator for the chemical entity furthercomprises: deriving fermionic Hamiltonian for the chemical entity, andtransforming fermionic Hamiltonian to a qubit Hamiltonian using one ormore transformation algorithms, wherein the one or more transformationalgorithm comprises any one or more of: Jordan-Wigner transformation,Parity transformation and Bravyi-Kitaev transformation. The methodfurther includes converting the derived evolution operator to anevolution operator for a quantum computer to be used for simulation ofthe evolution of the at least one quantum state according to theevolution operator and the one or more parameters; and generating aquantum circuit for simulation of the quantum system for the chemicalentity, which is optimized by performing one or more operations tominimize quantum resources to be used for the generated quantum circuitbased on the one or more parameters; and placing quantum resources amongone or more elementary logical units (ELUs) based on any one or more of:frequency of occurrence of the quantum resources in the generatedquantum circuit, order of occurrence of the quantum resources in thegenerated quantum circuit, connectivity parameters between one or morequantum resources, efficiency of gates between specific quantumresources, quality of gates between specific quantum resources or acombination thereof.

The computer-implemented method, system and computer-readable mediumstoring executable instructions for optimizing a quantum circuit aredescribed in detail below and illustrated in accompanying figures.

FIG. 2A and FIG. 2B illustrate at least some portions of an exemplarymethod for optimizing a quantum circuit, the method includes receivingone or more parameters for simulation of evolution of at least onequantum state of a chemical entity to be simulated via step 202. The oneor more parameters for simulation of evolution of the at least onequantum state of the chemical entity include optimization parameters,wherein the optimization parameters include any one or more of: an errortolerance for the quantum circuit, a number of gates permitted for thequantum circuit, a number of qubits permitted for the quantum circuit;and atomic information for the chemical entity including atomiccoordinates and atomic number etc. The molecular orbital calculationsare performed for the chemical entity chosen, e.g., conversion of SlaterType Orbitals to Gaussian Type Orbitals (STO-3G) is performed via step208 as depicted in FIG. 2A and via step 208′ as depicted in FIG. 2B.These calculations further involve deriving evolution operator e^(iHt)for the chemical entity from Schrödinger's Equation via step 204.

In an embodiment, the evolution operator for the chemical entity isderived from Schrödinger's Equation, which may be time dependent or timeindependent, where is a wave function for electron system for thechemical entity, His a Hamiltonian operator and E is energy of theelectron for the electron system for the chemical entity. The evolutionoperator e^(iHt) may be derived from time independent Schrödinger'sEquation: {umlaut over (H)}ψ=Eψ;

or time dependent Schrödinger's Equation:

${i\;\hslash\frac{\partial}{\partial t}\left. {\Psi\left( {r,t} \right)} \right\rangle} = {\hat{H}\left. {\Psi\left( {r,t} \right)} \right\rangle}$

The Hamiltonian coefficient or overlap integral calculated by the one ormore of the above-mentioned methods, e.g., using the Schrödinger'sEquation, for the chemical entity, is known as a fermionic Hamiltonian.For example, numerical values of the coefficient h_(pq) and h_(pqrs)based on the chosen chemical basis for a chemical entity, e.g., amolecule, may be calculated using an equation:

$= {{\sum\limits_{p,q}{h_{pq}f_{p}^{\dagger}f_{q}}} + {\sum\limits_{p,q,r,s}{h_{pqrs}f_{p}^{\dagger}f_{q}^{\dagger}f_{r}f_{s}}}}$

Wherein f^(\)≡ Creation operator, and f Annihilation operator; andwherein the first term represents kinetic energy plus electron-coreinteraction and the second term represents electron-electroninteraction; and wherein Overlap Integrals are represented as:

$h_{pq} = {\int{d\;\overset{\rightarrow}{r}\;{\chi_{p}^{*}\left( \overset{\rightarrow}{r} \right)}\left( {+ {\sum\limits_{nuc}{\hat{V}}_{nuc}}} \right){\chi_{q}\left( \overset{\rightarrow}{r} \right)}}}$$h_{pqrs} = {\int{d\;{\overset{\rightarrow}{r}}_{1}d\;{\overset{\rightarrow}{r}}_{2}\frac{{\chi_{p}^{*}\left( {\overset{\rightarrow}{r}}_{1} \right)}{\chi_{q}^{*}\left( {\overset{\rightarrow}{r}}_{2} \right)}{\chi_{r}\left( {\overset{\rightarrow}{r}}_{2} \right)}{\chi_{s}\left( {\overset{\rightarrow}{r}}_{1} \right)}}{r_{12}}}}$${\chi\left( \overset{\rightarrow}{r} \right)} \equiv {Orbital}$Slater  type  orbitals  (STO)− > Gaussian  type  orbitals  (GTO)

The numerical values of the coefficient h_(pq) and h_(pqrs) based on thechosen chemical basis thus computed are stored in a memory and usedfurther for conversion from natural basis to qubit basis. The processormay be provided with rules to decide which chemical basis to choosebased on atomic information, e.g., coordinates, atomic numbers, andoptimization parameters and hardware constraints, e.g., an errortolerance for the quantum circuit, a number of gates permitted for thequantum circuit, a number of qubits permitted for the quantum circuit,connectivity parameters between the qubits, speed, efficiency andquality of gates between specific pairs (or sets) of qubits in thesystem, etc. provided.

The generation of the optimized quantum circuit for simulation of thequantum system for the chemical entity, which may be an atom, a moleculeor an ion, includes choosing a chemical basis. Although embodiments ofthe system and process described herein are explained using an exemplarychemical basis STO-3G for hydrogen-like atom since it is the simplestminimal basis set, other chemical bases may also be chosen.

The chemical basis chosen comprises atomic orbitals, e.g., Gaussian-typeorbitals, Slater-type orbitals, or numerical atomic orbital. Thechemical entity for which the chemical basis is chosen may be as simpleas hydrogen atom (one basis function for minimal chemical basis) or maybe any other complex molecule or ion for which atomicinformation/parameters, e.g., coordinates, atomic numbers is known andmay be provided as user input. The user input may also include hardwareconstraints along with optimization parameters, e.g., error rate of eachquantum gate, error tolerance for the quantum circuit, number of qubitsavailable in the hardware, connectivity between the qubits, maximumnumber of gates permitted for the quantum circuit, maximum number ofqubits permitted for the quantum circuit, and maximum duration of timeallowed for the simulation, etc. Other constraints may also be provided.

The Hamiltonian of interest includes the Hamiltonian coefficients h's,which entail the information about the energy of the system written insome chemical basis, e.g., a hydrogenic-like slater-type orbital basis.If this orbital is then approximated using three Gaussian functions(STO-3G), Hamiltonian coefficients h may be explicitly calculated.Hydrogenic-like slater-type orbital basis is chosen here as an exampleonly and other bases may also be used, similarly, STO-3G is chosen hereas an example, since it is a minimal basis, however, other combinationsof functions may also be used, as described above.

The derived fermionic Hamiltonian thus obtained is then converted to aqubit Hamiltonian using one or more transformation algorithms via step206 as depicted in FIG. 2A and via step 210′ as depicted in FIG. 2B. Thetransformation algorithm used for such transformation may include anyone or more of: Jordan-Wigner transformation, Parity transformation andBravyi-Kitaev transformation. Other suitable transformation algorithms,or a combination of more than one transformation methods, may also beused. For example, Jordan-Wigner transformation, which may berepresented by the following equation:

$= {{{\sum\limits_{p,q}{h_{pq}f_{p}^{\dagger}f_{q}}} + {\sum\limits_{p,q,r,s}{h_{pqrs}f_{p}^{\dagger}f_{q}^{\dagger}f_{r}\left. f_{s}\downarrow \right.}}} = {{\sum\limits_{p,q}{h_{pq}a_{p}^{\dagger}a_{q}}} + {\sum\limits_{p,q,r,s}{h_{pqrs}a_{p}^{\dagger}a_{q}^{\dagger}a_{r}a_{s}}}}}$

The calculated and converted time evolution operator e^(iHt) is thenapproximated using Trotter formula for approximation via step 206 asdepicted in FIG. 2A and via step 212′ as depicted in FIG. 2B. Othersuitable methods for approximation may also be used, e.g., algorithmsbased on high-order product formulas (PFs), direct application of theTaylor series (TS), quantum signal processing method (QSP) etc. Thechoice of approximation parameters depends on optimization parametersincluding any one or more of: error tolerance for the quantum circuit,number of gates permitted for the quantum circuit, number of qubitspermitted for the quantum circuit etc. The optimization parameters mayalso include minimization of number of quantum resources used in thegenerated quantum circuit so as to minimize errors, as use of morequantum resources may induce more errors. The Trotter formula used toapproximate the time evolution of the quantum system, or used to createthe variational trial wavefunction (also known as the ansatz function)may be represented as follows:

${\exp\left( {{- {it}}\;{\sum\limits_{j = 1}^{L}{\alpha_{j}H_{j}}}} \right)} \approx \left\lbrack {S_{2k}(\lambda)} \right\rbrack^{r}$${S_{2}(\lambda)}:={\prod\limits_{j = 1}^{L}{{\exp\left( {\alpha_{j}H_{j}{\lambda/2}} \right)}{\prod\limits_{j = L}^{1}{\exp\left( {\alpha_{j}H_{j}{\lambda/2}} \right)}}}}$S_(2k)(λ) := [S_(2k − 2)(p_(k)λ)]²S_(2k − 2)((1 − 4p_(k))λ)[S_(2k − 2)(p_(k)λ)]²Wherein the first order formula may be represented as

e^(i(A + B)ɛ) = 1 + i(A + B)ϵ − (A² + AB + BA + B²)ϵ²/2 + …e^(i A ϵ)e^(i B ϵ) = 1 + i(A + B)ϵ − (A² + 2AB + B²)ϵ²/2 + … → Match  up  to  first  orderAnd 2k^(th) order formula may be represented as:

${\exp\left( {{- {il}}\;{\sum\limits_{j = 1}^{L}{\alpha_{j}H_{j}}}} \right)} = {{\exp\left( {\lambda{\sum\limits_{j = 1}^{L}{\alpha_{j}H_{j}}}} \right)}^{r} \approx {\prod\limits_{j = 1}^{L}\left\lbrack {\exp\left( {\alpha_{j}H_{j}\lambda} \right)} \right\rbrack^{r}}}$

The approximated evolution operator thus obtained and the calculatedHamiltonian coefficient for the chemical entity under consideration forthe simulation are then used as an input to the circuit synthesistoolkit to generate a generic circuit that simulates the quantum systemfor the chemical entity via step 214 and 216′ as depicted in FIGS. 2Aand 2B respectively. The generated quantum circuit is then optimized byusing one or more of: mapping and placement of quantum resources,parallel processing, scheduling, and using heuristic optimization, etc.and is further illustrated in FIGS. 3A, 3B, 4A, 4B, 4C, 4D, 4E and 4F,and description accompanying FIGS. 3A, 3B, 4A, 4B, 4C, 4E and 4F.

Quantum chemistry calculations are very complex and hence require alarge number of quantum resources for a quantum computer to performthose computations. As discussed above in the description accompanyingFIG. 1A and FIG. 1B, exemplary system architecture includes a number ofextended elementary logic units (EELUs). Each EELU further includes anumber of registers, also known as elementary logic units (ELUs), whichfurther include trapped ion qubits. Controlling the states of thesequbits by applying adequate control signal, for example, via laser beam,to one or more subsets of these qubits, quantum gates between the subsetof qubits may be realized via local Coulomb interactions between qubits.The qubits from different EELUs and ELUs are interconnected or entangledthrough photonic quantum channels using reconfigurable optical orphotonic switch.

Therefore, the resources may be arranged within different ELUs based onthe frequency of appearance due to different levels of connectivity forqubits that are within the same ELU; for qubits that are withindifferent ELUs but within the same EELU; and for qubits that are withindifferent EELUs. For example, the elementary logic units (ELUs) shown inFIG. 1A and FIG. 1B include a number of qubits. It is difficult to growthe ELUs beyond certain number of qubits, for example 30 qubits per ELUand still maintain connectivity among the qubits included in that ELU.One of the ways to overcome this issue is by creating more ELUs with acertain number of qubits, for example ten ELUs per one EELU, where eachELU includes, for example, 30 qubits, and providing connections betweendifferent ELUs. The number of qubits illustrated here are for exampleonly and a smaller or larger number of qubits per ELU may be used toallow for a technically feasible and efficient system. Also, the numberof qubits in each ELU and the number of ELUs in each EELU may not be allidentical: one can imagine ELUs with different numbers of qubits in eachELU, or EELUs with different number of ELUs in each EELU. Furthermore,the number of qubits in each ELU and the number of ELUs in each EELU canbe dynamically reconfigured by software control, e.g., the number ofquantum resources in each ELU, the number of ELUs in each EELU and thenumber of EELUs used in the computations can be varied over the durationof the computation. The gate operation between the qubits included inthe single ELU are simpler than the gate operations between the qubitsfrom different ELUs, since it involves moving a subset of qubits fromone ELU to another ELU. However, since the EELUs reside on a chip, thereare limitations as to the number of ELUs that can be accommodated withina single EELU. Therefore, a number of EELUs, each with a number ELUscontained within them, are manufactured. The photonic or optical switchnetwork shown in FIG. 1A allows the connection between one or morequbits from one EELU with one or more qubits from another EELU, formingpairs of entangled qubits between the EELUs, as required to perform thequantum circuit operations. The process of connecting a qubit from oneEELU with a qubit from another EELU is also known as teleportation (ofeither qubits or gates), which in this case is achieved via generatingshared entangled pairs of qubits, rather than physical movement ofqubits which is done when a qubit from one ELU is entangled with a qubitfrom another ELU within the same EELU.

Since the qubits within the ELU are very well connected, qubits fromdifferent ELUs on the same EELU are well connected and qubits fromdifferent EELUs are somewhat connected, it is important to arrange ormap the qubits in an efficient way to accommodate specific circuitblocks that show up in the quantum chemistry calculations based on thefrequency of appearance/occurrence of qubits within that circuit block.The placement of qubits may additionally or alternatively be based onthe sequence of appearance of qubit gates among them to accommodatespecific circuit blocks that show up in the quantum chemistrycalculations. The process of placing of specific qubits on specific ELUsbased on frequency of appearance and/or order of appearance can beautomated using a software tool, that takes into account the constraintsof the underlying quantum computer hardware.

Additionally, the qubits where maximum interaction is required toperform quantum operations may be placed close to each other, forexample, within a single ELU and close to each other within the singleELU, thereby saving gates utilized for connecting the two qubits thatare placed next to or close to each other. This may result in placementof other qubits far from each other, however, since connecting thequbits that are placed far from each other may be required fewer times,even though that connection may require more time or other resourceslike more quantum logic gates, it will still result in overall smallernumber of gates being used and thus utilization of less quantumresources or reduced execution time. The methods for optimization viaefficient mapping and placement of quantum resources using differenttechniques is described herein and also illustrated by example circuitsshown in FIGS. 3A, 3B, 4A, 4B, 4C, 4D, 4E and 4F, and described indetail in the description accompanying FIGS. 3A, 3B, 4A, 4B, 4C, 4D, 4Eand 4F.

Due to inherent limitation as to number of qubits that can beaccommodated within a single ELU and number of ELUs that can beaccommodated within a single EELU, additional optimization using othertechniques such as but not limited to scheduling as well as parallelprocessing and storing the results of the parallel processing may alsobe employed. For example, different computations may be performed ondifferent EELUs in parallel and the results may be stored until they arerequired for further processing. Quantum teleportation process isdiscussed in U.S. Pat. No. 9,858,531 and discussed above in thedescription accompanying FIGS. 1A and 1B, may be used to carry outcommunication between the parallelly processed computations. Forexample, the various terms in the Trotter formula can in practice bere-ordered and executed while keeping the errors within the allowedbound. If there are several terms in the Trotter formula that do notinvolve overlapping qubits, their order can be shifted without anypenalties. Therefore, each group of terms in the Trotter formula that donot share common qubits between the groups can be mapped on to a singleELU (or EELU), and the operations can be executed in parallel. Then,those terms that share common qubits between the groups can be executed,by exercising communication mechanism of qubits between ELUs (such asshuttling of ion qubits) or between EELUs (such as photonicinterconnect) which are connected by a qubit communication bus. Suchmethods for optimization via mapping and placement of qubits, parallelprocessing, gate scheduling and teleporting are illustrated by FIGS. 3A,3B, 4D, 4E and 4F and described in detail in the descriptionaccompanying FIGS. 3A, 3B, 4D, 4E and 4F.

The generic circuit generated for chosen chemical basis as illustratedin FIGS. 2A and 2B and described in the description accompanying FIGS.2A and 2B is thus mapped to generate quantum circuit via step 214 asillustrated in FIG. 2A and 214′ as illustrated in FIG. 2B where thequantum resources used in the quantum circuits are optimized orminimized via step 216 as illustrated in FIG. 2A and 216′ as illustratedin FIG. 2B. This optimization or minimization of quantum resources isfurther illustrated in FIGS. 4A, 4B and 4C and described in detail indescriptions accompanying FIGS. 4A, 4B and 4C.

The quantum circuit for simulation of the quantum system for thechemical entity is then further optimized by using different techniquesincluding any one or more of: mapping and placement of quantumresources, parallel processing, gate scheduling, heuristic optimization,etc. via step 218 as illustrated in FIG. 2A and 218′ as illustrated inFIG. 2B. This further optimization of quantum circuit using mapping andplacement of quantum resources, parallel processing and gate schedulingis further illustrated in FIGS. 3A, 3B, 4D, 4E and 4F and described indetail in descriptions accompanying FIGS. 3A, 3B, 4D, 4E and 4F.

The quantum resources for optimization of the quantum circuit mayinclude any one or more of: quantum computations used for simulationwhich may further include number of qubits used in a circuit block aswell as number of quantum gates, error tolerance for the quantum circuitand time taken to provide results of the simulation. The optimizedquantum circuit thus produced may provide time evolution of the chosenchemical entity which may not be limited to a short-term evolution, forexample, timescales corresponding to establishment of the steady-stateenergy spectrum, but rather over an extended period of time, such as thetimescales corresponding to propagation of an excitation above theground-state of the energy of the system, or the timescale to controlexternal parameters to force the time-evolution of the system (such asin adiabatic time evolution of a quantum system).

The core of circuit optimization lies in revealing the repeated patternof Pauli string prefix in the individual qubit-Hamiltonian termsrepresented in the Pauli-matrix basis that may arise from thetransformation of the physical-level Hamiltonian. For example, fortypical chemistry problems, the Wigner-Jordan transformation representsthe fermion creation operator {circumflex over (α)}_(j) and theannihilation operator {circumflex over (α)}_(j) ^(\) in the Pauli Matrixbasis are represented as:{circumflex over (α)}_(j)=

⊗^(j−1)⊗{circumflex over (σ)}⁻⊗{circumflex over (σ)}_(z) ^(N−j−1){circumflex over (α)}_(j) ^(\)=

⊗^(j−1)⊗{circumflex over (σ)}₊⊗{circumflex over (σ)}_(z) ^(N−j−1)

Using this transformation and noting that {circumflex over(σ)}_(±)={circumflex over (σ)}_(x)±i{circumflex over (σ)}_(y), the termsin the Hamiltonian of the chemical entity, h_(ji)â_(j) ^(\)â_(i) orh_(jirs)â_(j) ^(\)â_(j) ^(\)â_(r)â_(s), will be expressed as a productof Pauli operators. In general, the products of two or four σ_(±)'s leadto a two- or four-fold product of {circumflex over (σ)}_(x) and{circumflex over (σ)}_(y) operators corresponding to the electronoccupation in the molecular orbital represented by the qubit,accompanied by a series of {circumflex over (σ)}_(z) operators“connecting” the terms that involve the {circumflex over (σ)}_(x) and{circumflex over (σ)}_(y) operators to reflect the proper anti-symmetryof the electron-electron exchange. This series of {circumflex over(σ)}_(z) operators is often referred to as the Jordan-Wigner strings.

Thus, for solving chemistry problems by performing operations usingquantum circuit, connections between certain qubits may be required.FIGS. 3A and 3B show two different (yet equivalent) methods ofimplementing representative circuits corresponding to two terms of theHamiltonian in the Trotter expansion, for example, FIG. 3A shows oneterm 302 involving qubits 1, 2, 3 and 4, followed by another term 304involving qubits 4, 5, 6 and 7, similarly, FIG. 3B shows one term 302′involving qubits 1′, 2′, 3′ and 4′, followed by another term 304′involving qubits 4′, 5′, 6′ and 7′. Although 7 qubits are shown here asan example, the person skilled in the art may readily recognize that asmaller or larger number of qubits may be used.

The method shown FIG. 3A shows the implementation of the four-productPauli matrices using a series of controlled-NOT (CNOT) gates eachconnecting two qubits (where the target qubit, e.g., 306 denoted by ⊕ isflipped between |0

and |1

if and only if the control qubit, e.g., 308, denoted by the filled dot,is in the |1

state), followed by shifting the relative phase of the 4^(th) qubit bythe angle θ (denoted by R_(z)(θ) 310, where the phase of the |1

state is shifted by e^(iθ) relative to the |0

state), followed by the inverse sequence of the CNOTs applied earlier inthe sequence. In this circuit, the CNOT gates in the series are appliedbetween arbitrary pair of qubits in the system, and share a commontarget qubit 306. This is referred to as a “jump-type” implementation ofthe Hamiltonian evolution. In the presence of Jordan-Wigner strings, thesequence of the CNOT gates can be significantly longer.

FIG. 3B represents an alternative way of implementing the circuitcorresponding to the time evolution due to the same Hamiltonian term.The method shown FIG. 3B shows the implementation of the four-productPauli matrices using a series of controlled-NOT (CNOT) gates eachconnecting two qubits (where the target qubit, e.g., 306′ denoted by ⊕is flipped between |0

and |1

if and only if the control qubit, e.g., 308′, denoted by the filled dot,is in the |1

state), followed by shifting the relative phase of the 4^(th) qubit bythe angle θ (denoted by R_(z)(θ) 310′, where the phase of the |1

state is shifted by e^(iθ) relative to the |0

state), followed by the inverse sequence of the CNOTs applied earlier inthe sequence. In this method, the CNOT gate sequences are applied inseries between nearest-neighbor qubits. This is referred to as a“step-type” implementation of the Hamiltonian evolution.

Two identical CNOT gates applied back-to-back leaves the initial stateto be the same as the final state. In that sense, application of twoCNOT gates back-to-back is equivalent to doing nothing at all, and sotwo CNOT gates cancel each other out if they are applied back-to-back.Proper scheduling of gates may result in more instances of gatecancellations and hence minimization of quantum resources leading toerror minimization. Gate scheduling is a process by which the sequenceof quantum logic gates that constitute the quantum computation process,for example, for a chemistry calculation, is applied to the qubits oncethey are placed onto the physical qubits in the quantum computerhardware. The circuits shown in FIGS. 3A and 3B are examples ofscheduling of gates on qubits that are assigned or mapped to representthe electron populations for molecular orbits. Optimized gate schedulingcan lead to better circuit performance, due to gate cancellations and/orparallel processing.

For computing the properties of a complex chemical entity, the timeevolution due to a large number of Hamiltonian terms have to besimulated using the Trotter expansion. By carefully ordering theHamiltonian terms in the Trotter expansion, many of the Jordan-Wignerstrings can be cancelled out. The process of mapping of specific qubitsrepresenting the molecular orbitals in the Hamiltonian to physicalqubits in an ELU on the quantum computer is known as qubit mapping. Inorder to take advantage of the full connectivity within an ELU, theHamiltonian terms in the Trotter expansion may be divided into groups,where each group contains many of the qubits that are used repetitivelyin those terms, and assign them on to the physical qubits in a singleELU. The efficient mapping of specific qubits representing the molecularorbitals in the Hamiltonian to physical qubits and efficient placementof physical qubits (qubits) allows for the simulations in each group tobe started in parallel in each ELU or EELU. There inevitably are qubitsthat are shared between different groups: these qubits are placed in oneELU first, and once all the simulations involving that qubit iscompleted in the group, it may be either shuttled to another ELU in thesame EELU, or teleported to the proper ELU in another EELU, so thatfurther simulation involving this qubit in a different group maycontinue. The shuttling of qubits from one ELU to another ELU within thesame EELU provides a connection between the different ELUs of the sameEELU via a qubit communication bus. In order to minimize thecommunication costs between ELUs and EELUs, the placing of the qubits tothe ions in the ELUs are done in such a way that the number of shuttlingbetween ELUs within the same EELU and teleportation between EELUs areminimized. Thus, mapping and placing of quantum resources or qubitsamong one or more elementary logical units (ELUs) is based on any one ormore of: frequency of occurrence of the quantum resources in thegenerated quantum circuit, order of occurrence of the quantum resourcesin the generated quantum circuit, connectivity parameters between one ormore quantum resources, efficiency of gates between specific quantumresources, quality of gates between specific quantum resources, etc.

Various circuit optimization techniques may be used to reduce the numberof gates needed to implement the Hamiltonian evolution. FIGS. 4A, 4B and4C show example processing flows by which at least portions of themethod for optimizing a quantum circuit are implemented, in accordancewith at least some embodiments described herein, e.g., different stepsinvolved in Heuristic optimization. For example, FIG. 4A illustratesterm merging, wherein two Hamiltonian terms with identical Pauliproducts may be combined into a single term. FIG. 4B illustrates anexemplary qubit re-embedding technique: the two Hamiltonian terms inthis example, namely {circumflex over (σ)}_(z){circumflex over(σ)}_(z){circumflex over (σ)}_(z){circumflex over (σ)}_(z) and{circumflex over (σ)}_(x){circumflex over (σ)}_(z){circumflex over(σ)}_(z){circumflex over (σ)}_(z), utilize two similar “step-type”implementation of each Trotter term. Simply by labeling the qubitsdifferently so that the Hadamard (H) gate 401′ in between the two termsis applied to the same qubit on which the R_(z)(θ) gate is applied, fourCNOT gates 403 may be eliminated in the process. FIG. 4C illustrates anexemplary merge-sorting process, where the terms in the Trotterexpansion are ordered so that the number of basis changes among thePauli matrices (Î, {circumflex over (σ)}_(x), {circumflex over (σ)}_(y)and {circumflex over (σ)}_(z)) between the subsequent terms is minimized(usually reduced to one or two). In this case, many of the CNOT gates inthe implementation of each Trotter term cancels out between neighboringterms, using either the “jump-type” or “step-type” implementation of theterms. Using the “jump-type” implementation results in morecancellations of the CNOT gates than the “step-type” implementation whenthe basis for two of the four terms are changed between {circumflex over(σ)}_(x) and {circumflex over (σ)}_(y).

FIG. 4D shows an example processing flow by which at least portions ofthe method for optimizing a quantum circuit are implemented, inaccordance with at least some embodiments described herein. As discussedabove, due to inherent limitation as to number of qubits that can beaccommodated within a single ELU and number of ELUs that can beaccommodated within a single EELU, additional optimization using othertechniques such as but not limited to qubit mapping, placement ofqubits, gate scheduling as well as parallel processing and storing theresults of the parallel processing for later use may also be employed.For example, when using Trotter-Suzuki approximation where repetitivecalculations have to be performed, which may be represented as:e ^(i)(H ₀ +H ₁ +H ₂ . . . )t≅(e ^(iH) ⁰ t/ne ^(iH) ¹ ^(t/n) e ^(iH) ²^(t/n) e ^(iH) ³ ^(t/n) . . . )^(n)

FIG. 4D illustrates an example of using an extra qubit to parallelizethe circuit execution. In the example 410, a common qubit 416 is usedfor two different controlled operations, for example, controlled-U 412and controlled-V 414. While all other qubits associated with operation U412 and operation V 414 are independent, the sharing of common controlqubit 416 prohibits scheduling and executing the two operationsoperation U 412 and operation V 414 in parallel. In the example 420, anentangled pair, for example, 426 and 428, for operation U 422 and foroperation V 424 of control qubits, is created with the CNOT gate, andeach qubit in the entangled pair is allowed to control the operation U422 and operation V 424. At the end of the controlled-U 422 andcontrolled-V 424 process, another CNOT gate for the pair of controlqubits may be used to restore the initial state of the control qubits(not shown). Although the two figures are equivalent, in 410, the twocontrolled operations (controlled-U 412 and controlled-V 414) can bescheduled to run in parallel, as they involve completely independent setof qubits, that is do not share a common qubit.

FIG. 4E shows an example of parallel processing as well as schedulingthe gate execution, where different computations may be performed ondifferent EELUs, 444 and 446, in parallel and the results may be storeduntil they are required for further processing. In this example, onequbit must be shared between the quantum circuits carried out on EELU444 and the quantum circuit carried out on EELU 446. This shared qubitis first mapped to qubit 437 placed in EELU 444, and the unitaryoperation U 432 controlled by this qubit 437 is carried out. After that,the qubit 437 must be teleported to EELU 446 for further processing ofcontrolled-V operation 434 involving qubits in EELU 446. This isaccommodated by an entangled pair |ψ>, established between a qubit 450in EELU 444 and a qubit 448 in EELU 446, through the photonic network.This entangled pair can be established at any time before the qubitteleportation is needed. Upon completion of the controlled-U operation432 in EELU 444, the control qubit 437 is teleported to qubit 448 inEELU 446, through the teleportation process 460 shown in the dottedrectangle. This involves a CNOT gate operation between qubit 437 andqubit 450, followed by a Hadamard gate 452 applied to qubit 437. Then,these two qubits 437 and 450 are measured in the computational basis (in|0> and |1>). Based on the outcome of these two measurements, a singlequbit Pauli gate of either X 456 and/or Z 458 is applied to the qubit418. When these operations are completed, qubit 448 assumes the exactstate of qubit 437 before the teleportation process. Then, qubit 448 canbe used as the control qubit to operate the circuit V 434 to the inputqubits 440. A parallel operation may be performed in this case, forexample, as a controlled-W operation 435 involving qubits only in EELU446 (control qubit 439 and the input qubits 440) can be executed whilethe controlled-U operation 432 is executed in EELU 444.

Although the example circuit shown in FIG. 4E involves a teleportedqubit 437 that is used as a “control” qubit 448 in controlled-U 432 andcontrolled-V 434 operations, the teleported qubit can be used for anyfunction in a portion of a quantum circuit that spans the two or moreEELUs.

FIG. 4F illustrates an example of distributing quantum chemistrycircuits over two EELUs, that share a common qubit. FIG. 4F shows adetailed illustration of the circuit illustrated in FIG. 4E anddescribed in detail in the description accompanying FIG. 4E. The lastqubit 437 of the upper circuit 432 is teleported to the lower circuit434, and used, for example, in the final group of (e^(iH) ^(n) ^(t/n))in the simulation of the Trotter term.

Gate scheduling is a process by which the sequence of quantum logicgates that constitute the quantum computation process, for example, fora chemistry calculation, is applied to the qubits once they are placedonto the physical qubits in the quantum computer hardware. The circuitsshown in FIGS. 3A, 3B, 4B, 4D, 4E and 4F are examples of scheduling ofgates on qubits that are assigned or mapped to represent the electronpopulations for molecular orbits.

Further optimization of quantum resources and optimization of quantumcircuit generated using optimized or minimized quantum resources may beperformed in series and iteratively until a specific performance target,e.g., pre-defined maximum tolerance threshold or tolerance limit for theerror, or minimum gate count, is achieved.

A sample circuit using native approach with no merging, no re-embeddingand no sorting is illustrated in FIG. 5. As configured, this examplecircuit requires 120,160 CNOT gates and 11,716 R_(z)(θ) gates. The tablebelow shows an example of the reduction of the number of gates whenvarious circuit optimization techniques are applied to this circuit. Italso reduces the number of Hadamard gates and Phase (S) gates used inthe circuit. More improvements can be obtained by further optimizing thecircuit.

Native Approach: No merging, Merge only Approach: Merging GreedyApproach: Merging, re-embedding no re-embedding, no sorting but nore-embedding or sorting (first order mode) and sorting # of controlledquantum logic gates # of controlled quantum logic gates # of controlledquantum logic gates (CNOT): 120,160 (CNOT): 108,802 (CNOT): 30,558 # ofR_(Z)(θ) gates: 11,716 # of R_(Z)(θ) gates: 10,319 # of R_(Z)(θ) gates:10,319 # of Hadamard (H) gates: 86,380 # of Hadamard (H) gates: 76,812 #of Hadamard (H) gates: 14,380 # of Phase (S) gates: 42,048 # of Phase(S) gates: 38,760 # of Phase (S) gates: 6,864

FIG. 6 illustrates a quantum computing system used to provide quantumstate measurements for a chemical entity chosen for which the quantumcircuit is optimized using method and system for optimization of quantumcircuit according to one or more embodiments described herein. Forexample, variational quantum eigensolver (VQE) method, where a trialfunction for the quantum state (also known as the ansatz state) isprepared using the quantum circuit 602 utilizing the Trotter-likeapproach with a set of variational parameters. Once the state isprepared, it is used to estimate the expectation value of the energy(Hamiltonian). In this method, a minimum in the energy is sought bychanging the variational parameters, and preparing new trial statesutilizing these updated parameters. The total Hamiltonian is estimatedby measuring the expectation values 606 corresponding to each term thatconstitutes the overall Hamiltonian. Since a Hamiltonian for a chemicalentity includes a combination of Hamiltonian terms, the measurements areto be carried out in series, resulting in a number of iterations tomeasure individual terms in the Hamiltonian. One way to save quantumcost is to measure expectation value of more than one termssimultaneously. For example, for a quantum state with two qubits, theexpectation values for the operators

IZ

,

ZI

and

ZZ

can all be measured simultaneously from preparation of one trialfunction.

In an embodiment, a classical computing system for optimizing a quantumcircuit for simulation of the quantum system for the chemical entity isdisclosed as illustrated in FIG. 7 and described in detail thedescription accompanying FIG. 7. The system for optimizing a quantumcircuit comprising at least one processor and a memory wherein thememory stores executable instructions for optimizing a quantum circuitthat, upon execution by the processor, cause the processor to performfunctions including receiving one or more parameters for simulation ofevolution of at least one quantum state of a chemical entity to besimulated; generating a quantum circuit for simulation of evolution ofthe at least one quantum state of the chemical entity; performing one ormore operations to minimize quantum resources or qubits to be used forthe generated quantum circuit based on the one or more parameters; andplacing quantum resources among one or more elementary logical units(ELUs) based on any one or more of: frequency of occurrence of thequantum resources in the generated quantum circuit, order of occurrenceof the quantum resources in the generated quantum circuit, connectivityparameters between one or more quantum resources, efficiency of gatesbetween specific quantum resources, quality of gates between specificquantum resources or a combination thereof.

The system after receiving one or more parameters for simulation ofevolution of at least one quantum state of a chemical entity to besimulated, derives a time evolution operator e′t for the chemicalentity; converts the derived evolution operator to an evolution operatorfor a quantum computer to be used for simulation of the evolution of theat least one quantum state according to the evolution operator and theone or more parameters and generates a quantum circuit for simulation ofthe quantum system for the chemical entity, which is optimized byperforming one or more operations to minimize quantum resources orqubits to be used for the generated quantum circuit based on the one ormore parameters; and placing quantum resources among one or moreelementary logical units (ELUs) based on any one or more of: frequencyof occurrence of the quantum resources in the generated quantum circuit,order of occurrence of the quantum resources in the generated quantumcircuit, connectivity parameters between one or more quantum resources,efficiency of gates between specific quantum resources, quality of gatesbetween specific quantum resources or a combination thereof as describedherein. Further optimization of quantum resources and optimization ofquantum circuit generated using optimized or minimized quantum resourcesmay be performed in series and iteratively until a specific performancetarget, e.g., pre-defined maximum tolerance threshold or tolerance limitfor the error, or minimum gate count, is achieved. FIG. 7 illustratesone or more embodiments of the system used for optimizing a quantumcircuit for simulation of the quantum system for the chemical entityusing the method depicted in FIGS. 2A, 2B, 3A, 3B, 4A, 4B, 4C, 4D, 4Eand 4F.

Once the quantum resources are optimized, and the quantum circuitoptimized to use parallel processing and gate scheduling, the resultingoptimized quantum circuit may be further optimized by using a number ofiterations, also known as heuristic optimization, and the resultsobtained via quantum circuit are compared to the theoretical results anderror tolerance. This optimization is described in detail incommonly-owned, co-pending U.S. application Ser. No. 16/164,586,“AUTOMATED OPTIMIZATION OF LARGE-SCALE QUANTUM CIRCUITS WITH CONTINUOUSPARAMETERS”.

FIG. 7 shows an example system and an illustrative computing embodiment,in which any of the processes and sub-processes of quantum computationmay be implemented as computer-readable instructions stored on acomputer-readable medium. The computer-readable instructions may, forexample, be executed by a processor of a device, as referenced herein,having a network element and/or any other device corresponding thereto,particularly as applicable to the applications and/or programs describedabove for quantum circuit optimization.

In an embodiment, a computer-readable medium storing executableinstructions for optimizing a quantum circuit that, upon execution,cause a digital computing processor to perform functions includingreceiving one or more parameters for simulation of evolution of at leastone quantum state of a chemical entity to be simulated; generating aquantum circuit for simulation of evolution of the at least one quantumstate of the chemical entity; performing one or more operations tominimize quantum resources to be used for the generated quantum circuitbased on the one or more parameters; and placing quantum resources amongone or more elementary logical units (ELUs) based on any one or more of:frequency of occurrence of the quantum resources in the generatedquantum circuit, order of occurrence of the quantum resources in thegenerated quantum circuit, connectivity parameters between one or morequantum resources, efficiency of gates between specific quantumresources, quality of gates between specific quantum resources or acombination thereof.

In a very basic configuration, a computing device or a system 700 maytypically include, at least, one or more processors 702, a system memory704, one or more input components 706, one or more output components708, a display component 710, a computer-readable medium 712, and atransceiver 714.

Processor 702 may refer to, e.g., a microprocessor, a microcontroller, adigital signal processor, or any combination thereof.

Memory 704 may refer to, e.g., a volatile memory, non-volatile memory,or any combination thereof. Memory 704 may store, therein, operatingsystem 705, an application, and/or program data. That is, memory 704 maystore executable instructions to implement any of the functions oroperations described above and, therefore, memory 704 may be regarded asa computer-readable medium.

Input component 706 may refer to a built-in or communicatively coupledkeyboard, touch screen, or telecommunication device. Alternatively,input component 706 may include a microphone that is configured, incooperation with a voice-recognition program that may be stored inmemory 704, to receive voice commands from a user of computing device700. Further, input component 706, if not built-in to computing device700, may be communicatively coupled thereto via short-rangecommunication protocols including, but not limitation, radio frequencyor Bluetooth.

Output component 708 may refer to a component or module, built-in orremovable from computing device 700, that is configured to outputcommands and data to an external device.

Display component 710 may refer to, e.g., a solid state display that mayhave touch input capabilities. That is, display component 710 mayinclude capabilities that may be shared with or replace those of inputcomponent 706.

Computer-readable medium 712 may refer to a separable machine readablemedium that is configured to store one or more programs that embody anyof the functions or operations described above. That is,computer-readable medium 712, which may be received into or otherwiseconnected to a drive component of computing device 700, may storeexecutable instructions to implement any of the functions or operationsdescribed above. These instructions may be complimentary or otherwiseindependent of those stored by memory 704.

Transceiver 714 may refer to a network communication link for computingdevice 700, configured as a wired network or direct-wired connection.Alternatively, transceiver 714 may be configured as a wirelessconnection, e.g., radio frequency (RF), infrared, Bluetooth, and otherwireless protocols.

From the foregoing, it will be appreciated that various embodiments ofthe present disclosure have been described herein for purposes ofillustration, and that various modifications may be made withoutdeparting from the scope and spirit of the present disclosure.Accordingly, the various embodiments disclosed herein are not intendedto be limiting, with the true scope and spirit being indicated by thefollowing claims.

We claim:
 1. A computer-readable medium storing executable instructionsfor optimizing a quantum circuit that, upon execution, cause a digitalcomputing processor to perform functions comprising: receiving one ormore parameters for simulation of evolution of at least one quantumstate of a chemical entity to be simulated; generating a quantum circuitfor simulation of evolution of the at least one quantum state of thechemical entity; performing one or more operations to minimize quantumresources to be used for the generated quantum circuit based on the oneor more parameters; and placing quantum resources among one or moreelementary logical units (ELUs) based on any one or more of: frequencyof occurrence of the quantum resources in the generated quantum circuit,order of occurrence of the quantum resources in the generated quantumcircuit, connectivity parameters between one or more quantum resources,efficiency of gates between specific quantum resources, quality of gatesbetween specific quantum resources or a combination thereof, wherein theone or more elementary logical units (ELUs) are part of an ion-trapquantum computer.
 2. The computer-readable medium of claim 1, whereinthe chemical entity includes any one or more of: atoms, molecules, ionsand subatomic particles, wherein the subatomic particles are free, boundor localized.
 3. The computer-readable medium of claim 1, whereingenerating a quantum circuit for simulation of evolution of the at leastone quantum state of the chemical entity comprises: deriving evolutionoperator for the chemical entity to be utilized for the simulation usinga quantum computer, wherein deriving evolution operator for the chemicalentity further comprises: deriving fermionic Hamiltonian for thechemical entity, and transforming fermionic Hamiltonian to a qubitHamiltonian using one or more transformation algorithms, wherein the oneor more transformation algorithm comprises any one or more of:Jordan-Wigner transformation, Parity transformation and Bravyi-Kitaevtransformation.
 4. The computer-readable medium of claim 1, wherein theone or more operations to minimize quantum resources include any one ormore of term merging, re-embedding quantum resources and merge-sorting.5. The computer-readable medium of claim 1, wherein the one or moreparameters for simulation of evolution of at least one quantum state ofa chemical entity to be simulated include any one or more of: atomicinformation for the chemical entity, error tolerance for the quantumcircuit, maximum number of gates permitted for the quantum circuit,maximum number of qubits permitted for the quantum circuit, and maximumduration of time allowed for the simulation.
 6. The computer-readablemedium of claim 1, wherein optimizing the generated quantum circuitbased on the one or more parameters further comprises any one or moreof: gate scheduling and parallel processing of quantum computations. 7.The computer-readable medium of claim 6, wherein the gate scheduling forquantum computations is achieved by connecting one or more quantumresources from different ELUs, wherein the quantum resources areconnected by a qubit communication bus including shuttling of thequantum resources.
 8. The computer-readable medium of claim 6, whereinthe gate scheduling and parallel processing of quantum computations isachieved by connecting one or more quantum resources from differentextended elementary logical units (EELUs) via optical cross connect togenerate at least one entangled ion pair between a pair of EELUs.
 9. Thecomputer-readable medium of claim 8, wherein the number of quantumresources in each ELU, the number of ELUs in each EELU and the number ofEELUs used in the computations can be dynamically varied over theduration of the computation.
 10. The computer-readable medium of claim1, wherein the one or more ELUs are formed by jump-type implementationor step-type implementation of the Hamiltonian evolution.
 11. Thecomputer-readable medium of claim 8, wherein the extended EELUs are partof the ion-trap quantum computer.
 12. The computer-readable medium ofclaim 1, wherein one or more global gates are used to further minimizequantum resources to be used for the generated quantum circuit based onthe one or more parameters.
 13. The computer-readable medium of claim12, wherein the one or more global gates to further minimize quantumresources are provided by an ion-trap quantum computer.
 14. Acomputer-implemented method for optimizing a quantum circuit comprising:receiving one or more parameters for simulation of evolution of at leastone quantum state of a chemical entity to be simulated; generating aquantum circuit for simulation of evolution of the at least one quantumstate of the chemical entity; performing one or more operations tominimize quantum resources to be used for the generated quantum circuitbased on the one or more parameters; and mapping quantum resources amongone or more elementary logical units (ELUs) based on any one or more of:frequency of occurrence of the quantum resources in the generatedquantum circuit, order of occurrence of the quantum resources in thegenerated quantum circuit, connectivity parameters between one or morequantum resources, efficiency of gates between specific quantumresources, quality of gates between specific quantum resources or acombination thereof, wherein the one or more elementary logical units(ELUs) are part of an ion-trap quantum computer.
 15. Thecomputer-implemented method of claim 14, wherein the chemical entityincludes any one or more of: atoms, molecules, ions and subatomicparticles, wherein the subatomic particles are free, bound or localized.16. The computer-implemented method of claim 14, wherein generating aquantum circuit for simulation of evolution of the at least one quantumstate of the chemical entity comprises: deriving evolution operator forthe chemical entity to be utilized for the simulation using a quantumcomputer, wherein deriving evolution operator for the chemical entityfurther comprises: deriving fermionic Hamiltonian for the chemicalentity, and transforming fermionic Hamiltonian to a qubit Hamiltonianusing one or more transformation algorithms, wherein the one or moretransformation algorithm comprises any one or more of: Jordan-Wignertransformation, Parity transformation and Bravyi-Kitaev transformation.17. The computer-implemented method of claim 14, wherein the one or moreoperations to minimize quantum resources include any one or more of termmerging, re-embedding quantum resources and merge-sorting.
 18. Thecomputer-implemented method of claim 14, wherein the one or moreparameters for minimizing quantum resources to be used for the generatedquantum circuit include any one or more of: atomic information for thechemical entity, error tolerance for the quantum circuit, maximum numberof gates permitted for the quantum circuit, maximum number of qubitspermitted for the quantum circuit, and maximum duration of time allowedfor the simulation.
 19. The computer-implemented method of claim 14,wherein optimizing the generated quantum circuit based on the one ormore parameters further comprises any one or more of: gate schedulingand parallel processing of quantum computations.
 20. Thecomputer-implemented method of claim 19, wherein the gate scheduling forquantum computations is achieved by connecting one or more quantumresources from different ELUs, wherein the quantum resources areconnected by a qubit communication bus including shuttling of thequantum resources.
 21. The computer-implemented method of claim 19,wherein the gate scheduling and parallel processing of quantumcomputations is achieved by connecting one or more quantum resourcesfrom different extended elementary logical units (EELUs) via opticalcross connect to generate at least one entangled ion pair between a pairof EELUs.
 22. The computer-implemented method of claim 21, wherein thenumber of quantum resources in each ELU, the number of ELUs in each EELUand the number of EELUs used in the computations can be varied over theduration of the computation.
 23. The computer-implemented method ofclaim 14, wherein the one or more ELUs are formed by jump-typeimplementation or step-type implementation of the Hamiltonian evolution.24. The computer-implemented method of claim 21, wherein the extendedEELUs are part of the ion-trap quantum computer.
 25. Thecomputer-implemented method of claim 14, wherein one or more globalgates are used to further minimize quantum resources to be used for thegenerated quantum circuit based on the one or more parameters.
 26. Thecomputer-implemented method of claim 25, wherein the one or more globalgates to further minimize quantum resources are provided by an ion-trapquantum.
 27. A system for optimizing a quantum circuit comprising atleast one processor and a memory wherein the memory stores executableinstructions for optimizing a quantum circuit that, upon execution bythe processor, cause the processor to perform functions comprising:receiving one or more parameters for simulation of evolution of at leastone quantum state of a chemical entity to be simulated; generating aquantum circuit for simulation of evolution of the at least one quantumstate of the chemical entity; performing one or more operations tominimize quantum resources to be used for the generated quantum circuitbased on the one or more parameters; and placing quantum resources amongone or more elementary logical units (ELUs) based on any one or more of:frequency of occurrence of the quantum resources in the generatedquantum circuit, order of occurrence of the quantum resources in thegenerated quantum circuit, connectivity parameters between one or morequantum resources, efficiency of gates between specific quantumresources, quality of gates between specific quantum resources or acombination thereof, wherein the one or more elementary logical units(ELUs) are part of an ion-trap quantum computer.
 28. The system of claim27, wherein the chemical entity includes any one or more of: atoms,molecules, ions and subatomic particles, wherein the subatomic particlesare free, bound or localized.
 29. The system of claim 27, whereingenerating a quantum circuit for simulation of evolution of the at leastone quantum state of the chemical entity comprises: deriving evolutionoperator for the chemical entity to be utilized for the simulation usinga quantum computer, wherein deriving evolution operator for the chemicalentity further comprises: deriving fermionic Hamiltonian for thechemical entity, and transforming fermionic Hamiltonian to a qubitHamiltonian using one or more transformation algorithms, wherein the oneor more transformation algorithm comprises any one or more of:Jordan-Wigner transformation, Parity transformation and Bravyi-Kitaevtransformation.
 30. The system of claim 27, wherein the one or moreoperations to minimize quantum resources include any one or more of termmerging, re-embedding quantum resources and merge-sorting.
 31. Thesystem of claim 27, wherein the one or more parameters for minimizingquantum resources to be used for the generated quantum circuit includeany one or more of: atomic information for the chemical entity, errortolerance for the quantum circuit, maximum number of gates permitted forthe quantum circuit, maximum number of qubits permitted for the quantumcircuit, and maximum duration of time allowed for the simulation. 32.The system of claim 27, wherein optimizing the generated quantum circuitbased on the one or more parameters further comprises any one or moreof: gate scheduling and parallel processing of quantum computations. 33.The system of claim 32, wherein the gate scheduling for quantumcomputations is achieved by connecting one or more quantum resourcesfrom different ELUs, wherein the quantum resources are connected by aqubit communication bus including shuttling of the quantum resources.34. The system of claim 32, wherein the gate scheduling and parallelprocessing of quantum computations is achieved by connecting one or morequantum resources from different extended elementary logical units(EELUs) via optical cross connect to generate at least one entangled ionpair between a pair of EELUs.
 35. The system of claim 34, wherein thenumber of quantum resources in each ELU, the number of ELUs in each EELUand the number of EELUs used in the computations can be varied over theduration of the computation.
 36. The system of claim 27, wherein the oneor more ELUs are formed by jump-type implementation or step-typeimplementation of the Hamiltonian evolution.
 37. The system of claim 34,wherein the extended EELUs are part of the ion-trap quantum computer.38. The system of claim 27, wherein one or more global gates are used tofurther minimize quantum resources to be used for the generated quantumcircuit based on the one or more parameters.
 39. The system of claim 38,wherein the one or more global gates to further minimize quantumresources are provided by an ion-trap quantum computer.